Polarization
Two crossed polarizers let nothing through. Put a third one between them and 12.5% gets through, and adding a filter increases what passes because a polarizer does not only filter: it reorients.
01The direction in which the field oscillates
The fact that the electromagnetic wave is transverse leaves a freedom that longitudinal waves do not have: the electric field can oscillate in any direction in the plane perpendicular to the propagation. That direction is the polarization, and sound does not have it because it oscillates in the same direction in which it travels.
| State | What the field does | How it is achieved |
|---|---|---|
| Linear | Always oscillates in the same plane | A polarizer, or a single-element antenna |
| Circular | Rotates, describing a circle | Two equal components shifted by 90° |
| Elliptical | Rotates, describing an ellipse | The general case: any amplitudes or phase shift |
| Unpolarized | Changes randomly all the time | Any thermal source: the Sun, a lamp |
The last row is the natural state of almost all light: a lamp emits photons from atoms that are not coordinated with each other, and the result averages over all directions. Polarizing it is filtering, and that is why it always costs energy.
In satellite communications circular polarization is used because the relative orientation between the satellite and the ground antenna changes, and with linear polarization it would have to be tracked. Also a right-hand circular wave and a left-hand one are independent: two channels can be transmitted on the same frequency. And in the ionosphere, Faraday rotation turns the plane of a linear wave unpredictably, something that does not affect a circular one.
02Malus’s law
A polarizer lets through only the component of the field parallel to its axis. Since the intensity goes with the square of the field, the fraction that passes is:
Three polarizers with adjustable axes. Start with the first and the third crossed at 90°: nothing gets through. Then insert the middle one at 45° and see what happens.
It seems contradictory that adding a filter could increase what gets through, and it is not. The middle polarizer does not merely filter: it reorients. What comes out of it is polarized at 45°, and that direction does have a component along the axis of the third. Without it, what reaches the third is perpendicular to its axis and has none. It is the cleanest demonstration that a polarizer projects a vector, rather than simply letting through “the light that was already there.”
03Polarizing without a polarizer
Reflection polarizes. When light strikes a transparent surface at the Brewster angle, what is reflected is completely polarized in the plane parallel to the surface:
| Phenomenon | What it does | Where it appears |
|---|---|---|
| Brewster reflection | Polarizes what is reflected | Polarized sunglasses, laser windows |
| Rayleigh scattering | Polarizes the sky at 90° from the Sun | Photography filters; orientation of some insects |
| Birefringence | Two indices depending on the polarization | Quarter-wave plates, calcite |
| Optical activity | Rotates the plane of polarization | Measuring sugar concentration |
An LCD is pure polarization. Behind it there is a white lamp, in front two crossed polarizers —which by themselves would give black— and in between a layer of liquid crystal whose molecules rotate the plane of polarization. With no voltage, the layer rotates it 90° and the light passes through both polarizers: pixel on. With voltage, the molecules align, stop rotating anything and the second polarizer blocks: pixel off. Each pixel is a Malus experiment, and that is why the screen of a laptop goes dark when viewed through rotated polarized sunglasses.
04In the lab
With two polarizers, a stable source and a photodiode with an amplifier, measure the intensity every ten degrees of rotation of the second. Plot it against \( \cos^2\theta \) and verify that it gives a straight line.
Cross two polarizers until extinction and measure. Insert a third between them and sweep it from 0 to 90°, measuring. Locate the maximum and compare it with the 12.5 % predicted by theory.
Reflect a beam off a glass plate while varying the angle and observe the reflection through a polarizer. Find the extinction angle and from it obtain the refractive index of the glass.
05Common mistakes
- Believing that a polarizer “lets through the light that was already in its direction.” It projects: it transmits the component.
- Expecting the first one to let through more than half. With unpolarized light it is always 50 %, in any orientation.
- Thinking that adding a filter cannot increase what gets through. The middle one proves otherwise.
- Using Malus with unpolarized light. Malus holds from the second polarizer onward.
- Confusing circular polarization with unpolarized light. Circular is perfectly defined; it just rotates.
- Forgetting that polarization is transverse. Sound is not polarized: it oscillates in the direction in which it travels.
06Self-assessment
How much light passes through the first polarizer?
Half, if it enters unpolarized, whatever its orientation.
What fraction passes through three polarizers at 0°, 45° and 90°?
One eighth: \( \tfrac12 \times \cos^2 45° \times \cos^2 45° = 12.5\ \% \) of the light that entered.
Why does the middle one let light through if the other two are crossed?
Because it reorients: what comes out of it is at 45°, and that direction does have a component along the axis of the third.
What is the Brewster angle?
The angle of incidence at which the reflected light is totally polarized. Its tangent is the ratio of the indices.
Why do satellites use circular polarization?
Because it does not depend on the relative orientation, and the ionosphere rotates the plane of a linear wave unpredictably.
How does an LCD pixel work?
Two crossed polarizers with liquid crystal in between: with no voltage it rotates the polarization 90° and light passes; with voltage it does not rotate it and the second polarizer blocks it.
07Further reading
- Eugene Hecht. Optics. 5th ed. (Spanish edition, “Óptica”), Pearson, 2017. The whole of chapter 8: states of polarization, Malus, Brewster, birefringence and retarder plates.
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage, 2019. The introductory treatment, with the exercises on cascaded polarizers.
- Ernst Lueder. Liquid Crystal Displays: Addressing Schemes and Electro-Optical Effects. 2nd ed., Wiley, 2010. How a display is built from two polarizers and a layer of liquid crystal.